This learned divine was born in Swallow-street, Westminster, in 1731, where his father was in the humble employment of a stable-keeper. He was educated, however, at Westminster school at the same time with Smith and Vincent, who were afterwards his successors in the headship of that celebrated academy. In 1750 he was elected to Trinity college, Cambridge, where he took his bachelor's degree in 1754, and about the same time became usher of Westminster school. He then took orders, and officiated as morning-preacher of South Audley-street chapel. He continued in these employments — taking his master's degree in 1757 — until 1760, when he travelled into Germany, Italy, and France with Mr. Crewes afterwards member of parliament for Cheshire, who, when returned from his tour, settled on Dr. Hinchliffe £300 a year, and made him his domestic chaplain.
During his residence in Italy, he was favoured with an introduction to the duke of Grafton, who had been contemporary with him at Cambridge, and soon after, in 1764, by the interest of his grace, he was appointed head master of Westminster school, on the resignation of Dr. Markham, archbishop of York; but his ill state of health not being suited to such a laborious employment, he was obliged to resign in a few months after he had accepted it. He declined several advantageous offers that were made him if he would travel again; and being very easy in circumstances by the generosity of his friend and pupil, he intended to return and reside at college, when he was solicited by his noble patron to undertake for a few years the office of tutor to the young duke of Devonshire. In consequence of this, Dr. Hinchliffe removed to Devonshire-house, and remained there till his grace went abroad. By the joint interest of his two noble patrons, he was presented to the vicarage of Greenwich in 1766. Dr. Hinchliffe, it is said, was offered the tuition of the prince of Wales, which important trust he declined, from his predilection, as it is supposed, to what were called whig principles.
On the death of Dr. Smith, in 1768, his lordship was elected, through the recommendation of the duke of Grafton, master of Trinity college, Cambridge; and scarce a year had elapsed, when he was raised to the bishopric of Peterborough on the death of Dr. Lamb, in 1769, by the interest of the duke of Grafton, then prime minister. It is probable his lordship might have obtained other preferment, had he not uniformly joined the party in parliament who opposed the principle and conduct of the American war. The only other change he experienced was that of being appointed dean of Durham, by which he was removed from the mastership of Trinity college. He died at his palace at Peterborough, January 11th, 1794, after a long illness, which terminated in a paralytic stroke.
His lordship, although a man of considerable learning, published only three sermons, preached on public occasions. He was a graceful orator in parliament, and much admired in the pulpit. Mr. Jones, in his life of Bishop Horne, says that "he spake with the accents of a man of sense — such as he really was in a superior degree — but it was remarkable, and, to those who did not know the cause, mysterious, that there was not a corner in the church in which he could not be heard distinctly." The reason Mr. Jones assigns, was, that he made it an invariable rule "to do justice to every consonant, knowing that the vowels will be sure to speak for themselves. And thus he became the surest and clearest of speakers: his elocution was perfect, and never disappointed his audience." Two years after his death, a volume of Bishop Hinchliffe's sermons were published; but, probably from a want of judgment in the selection, did not answer the expectations of those who had been accustomed to admire him in the pulpit.